Problem 66 - Introductory Algebra for College Students

Question

Name the property used to go from step to step each time that “(why?)” occurs. $$\begin{array}{l} 5(x+4)+3 x \\ =(5 x+20)+3 x \text { (why?) } \\ =(20+5 x)+3 x \text { (why?) } \\ =20+(5 x+3 x) \text { (why?) } \\ =20+(5+3) x \text { (why?) } \\ =20+8 x \\ =8 x+20 \text { (why?) } \end{array}$$

Step-by-Step Solution

Answer

The properties used in each step are, in order: Distributive Property, Commutative Property, Associative Property, Distributive Property, Arithmetic operation, and Commutative Property.

Step 1: Identify the Property for First Change

In the first change, the expression inside the brackets in the term 5(x+4) is expanded to form 5x+20. This is because of the Distributive Property, which allows the 5 to be 'distributed' across the terms inside the brackets, resulting in \(5x + 20\).

Step 2: Identify the Property for Second Change

In the second transition from \(5x + 20\) to \(20 + 5x\), the Commutative Property is applied, which allows the addition operands to be reordered without changing the sum.

Step 3: Identify the Property for Third Change

The third transition, \(5x + 20 + 3x\) to \(20 + 5x + 3x\), uses the Associative Property of Addition that permits the regrouping of addition terms without affecting the result.

Step 4: Identify the Property for Fourth Change

In the fourth step, \(5x + 3x\) becomes \((5 + 3)x\) because of the Distributive Property. This 'factorization' they put \(x\) common and sum the multipliers.

Step 5: Identify the Property for Fifth Change

This step just computes \(5+3\) to give \(8x\), it's an Arithmetic operation.

Step 6: Identify the Property for Final Change

Finally, in the transition from \(20 + 8x\) to \(8x + 20\), the Commutative Property is again used, allowing the operands of the addition operation to be reordered.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to multiply a single term across terms in parentheses. In the expression \(5(x+4)\), this property lets us expand it to \(5x + 20\) by distributing the 5 across each term inside the parentheses. This means you multiply 5 by both \(x\) and 4, resulting in \(5x + 20\).
Simple, right? This property can be summarized as \(a(b+c) = ab + ac\). It is especially useful when simplifying expressions and equations.

• Helps in breaking down complex expressions.
• Useful for solving equations.
• Makes factoring easier.

Understanding the distributive property allows you to work with expressions more flexibly and accurately, forming a basis for more complex algebraic manipulations.

Commutative Property

The commutative property pertains to both addition and multiplication, allowing the terms of an operation to be swapped without altering the result. When you see \(5x + 20\) turned into \(20 + 5x\), we're invoking this property. The cultural commutative flavor means that it doesn't matter the order when you sum or multiply:

• For addition: \(a + b = b + a\)
• For multiplication: \(a \cdot b = b \cdot a\)

This property simplifies rearranging terms to perhaps make solving easier or matching terms visually more convenient.
Commutative property is quite helpful in mental calculations, allowing flexibility as you approach simplifying equations and working on computations.

Associative Property

The associative property deals with the grouping of numbers, indicating that how you group numbers in addition or multiplication does not change their sum or product. When transitioning from \((5x + 20) + 3x\) to \(20 + (5x + 3x)\), it’s easy to see this property in action. You are changing the grouping of terms.
Remember:

• For addition: \((a + b) + c = a + (b + c)\)
• For multiplication: \((a \cdot b) \cdot c = a \cdot (b \cdot c)\)

This concept is critical in algebra for regrouping terms to simplify equations and computations efficiently.
The associative property ensures that computations remain manageable, helping you focus on the actual operations without worrying about parentheses altering mathematical results.

Arithmetic Operations

Arithmetic operations are basic calculations such as addition, subtraction, multiplication, and division. In the original exercise, arithmetic operations are essential when calculating \(5 + 3\) to get \(8x\).

• Addition: Bringing quantities together.
• Subtraction: Determining the difference between quantities.
• Multiplication: Scaling one quantity by another.
• Division: Distributing a quantity into equal parts.

These operations form the bedrock of all mathematical calculations, providing the tools to perform more complex algebraic manipulations. Understanding arithmetic is crucial for simplifying expressions and solving algebraic equations, ensuring precision and efficiency in computation.
Mastering arithmetic operations enhances confidence and proficiency in handling calculations across various mathematical contexts.